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Periodic solutions of Hamiltonian systems of 3-body type. (English) Zbl 0745.34034
The existence of $$T$$-periodic solutions of Hamiltonian systems of the form $$\ddot q+V_ q(t,v)=0$$ is studied, where $$V=\sum^ 3_{1\leq i\neq j}V_{ij}(t,q_ i-q_ 2)$$ is of 3-body type, $$V(t,\xi)$$ is $$T$$- periodic in $$t$$ and singular at $$\xi=0$$. Under these hypotheses, provided that the singularity of $$V$$ at 0 is strong enough, it is proved that the functional $$\int^ T_ 0\left({1\over 2}|\dot q|^ 2- V(t,q)\right)dt$$ has an unbounded sequence of critical points, which correspond to the $$T$$-periodic solutions of the considered equation.
Reviewer: I.Ginchev (Varna)

##### MSC:
 34C25 Periodic solutions to ordinary differential equations 37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems 70F07 Three-body problems 58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces
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##### References:
 [1] Poincaré, H., LES méthodes nouvelles de la mécanique céleste, (1987), Libr. Albert Blanchard Paris · Zbl 0651.70002 [2] Bahri, A.; Rabinowitz, P. H., A minimax method for a class of Hamiltonian systems with singular potentials, J. Functional Anal., Vol. 82, 412-428, (1989) · Zbl 0681.70018 [3] A. Ambrosetti and V. Coti-Zelati, Critical Points with Lack of Compactness and Applications to Singular Hamiltonian Systems (to appear). · Zbl 0642.58017 [4] Degiovanni, M.; Giannoni, F.; Marino, A., Periodic solutions of dynamical systems with newtonian type potentials, in periodic solutions of Hamiltonian systems and related topics, (Rabinowitz, P. H.; etal., Vol. 29, (1987), NATO ASI Series Reidei, Dordrecht), 111-115 [5] Gordon, W. B., Conservative dynamical systems involving strong forces, Trans. Am. Math. Soc., Vol. 204, 113-135, (1975) · Zbl 0276.58005 [6] Greco, C., Periodic solutions of a class of singular Hamiltonian systems, Nonlinear Analysis: TMA, Vol. 12, 259-270, (1988) · Zbl 0648.34048 [7] Marino, A.; Prodi, G., Metodi perturbativi nella teoria di Morse, Boll. Un. Mat. Ital., Vol. 11, 1-32, (1975) · Zbl 0311.58006 [8] Bahri, A., Thèse de doctorat d’état, (1981), Univ. P. and M. Curie Paris [9] Bahri, A.; Berestycki, H., Forced vibrations of superquadratic Hamiltonian systems, Acta Math., Vol. 152, 143-197, (1984) · Zbl 0592.70027 [10] Borsuk, Shape Theory. [11] Sullivan, D.; Vigué-Poirier, M., The homology theory of the closed geodesic problem, J. Diff. Geom., Vol. 11, 633-644, (1976) · Zbl 0361.53058 [12] Dold, A., Lectures on algebraic topology, (1972), Springer-Verlag Heidelberg · Zbl 0234.55001 [13] P. H. Rabinowitz, Periodic Solutions for Some Forced Singular Hamiltonian Systems, (to appear), Festschift in honor of Jürgen Moser. [14] C. C. Conley, Isolated Invariant Sets and the Morse Index, C.B.M.S. Regional Conference Series in Math, #38, Am. Math. Soc., Providence R. I., 1978. [15] A. Bahri, (to appear). [16] Hirsch, M. W., Differential topology, (1976), Springer-Verlag · Zbl 0121.18004 [17] Spanier, E., Algebraic topology, (1966), McGraw-Hill · Zbl 0145.43303 [18] Klingenberg, W., Lectures on closed goedesics, (1978), Springer-Verlag [19] Ekeland, I., Une théorie de Morse pour LES systèmes hamiltoniens convexes, Ann. Inst. H. Poincaré: Analyse non linéaire, Vol. 1, 19-78, (1984) · Zbl 0537.58018 [20] A. Bahri and B. M. D’Oonofrio, (to appear).
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